Conservation of energy for schemes applied to the propagation of shallow-water inertia-gravity waves in regions with varying depth

The linear equations governing the propagation of inertia-gravity waves in geophysical fluid flows are discretized on the Arakawa C-grid using centere d differences in space. In contrast to the constant depth case it is demons trated that varying depth may give rise to increasing energy land loss of s tability) using the natural approximations for the Coriolis terms found in many well-known codes;This is true no matter which numerical method is used to propagate the equations. By a simple trick based on a modified weightin g that ensures that the propagation matrices for the spatially discretized equations become similar to skew-symmetric matrices, this problem is remove d and the energy is conserved in regions with varying depth too. We give a number of examples both of model problems and large-scale problems in order to illustrate this behaviour. In real applications diffusion, explicit thr ough frictional terms or implicit through numerical diffusion, is introduce d both for physical reasons, but often also in order to stabilize the numer ical experiments. The growing modes associated with varying depth, the C-gr id and equal weighting may force us to enhance the diffusion more than we w ould like from physical considerations. The modified weighting offers a sim ple solution to this problem. Copyright (C) 2000 John Wiley & Sons, Ltd.